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| ${\rm SO}(n)$-Invariant Special Lagrangian Submanifolds of ${\mathbb C}^{n+1}$ with Fixed Loci |
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Citation： |
Robert L. BRYANT.${\rm SO}(n)$-Invariant Special Lagrangian Submanifolds of ${\mathbb C}^{n+1}$ with Fixed Loci[J].Chinese Annals of Mathematics B,2006,27(1):95~112 |
| Page view： 1242
Net amount： 1026 |
Authors： |
Robert L. BRYANT; |
Foundation： |
Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical
Sciences Research Institute, and Columbia University. |
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| Abstract： |
Let~${\rm SO}(n)$ act in the standard way on~${\mathbb C}^n$ and
extend this action in the usual way to~${\mathbb C}^{n+1} =
{\mathbb C}\oplus{\mathbb C}^n$.
It is shown that a nonsingular special Lagrangian
submanifold~$L\subset{\mathbb C}^{n+1}$ that is invariant under
this~${\rm SO}(n)$-action intersects the fixed~${\mathbb C}
\subset{\mathbb C}^{n+1}$ in a nonsingular real-analytic arc~$A$
(which may be empty). If~$n>2$, then $A$ has no compact component.
Conversely, an embedded, noncompact nonsingular real-analytic
arc~$A\subset{\mathbb C}$ lies in an embedded nonsingular special
Lagrangian submanifold that is ${\rm SO}(n)$-invariant. The same
existence result holds for compact~$A$ if~$n=2$. If~$A$ is
connected, there exist $n$ distinct nonsingular ${\rm
SO}(n)$-invariant special Lagrangian extensions of~$A$ such that
any embedded nonsingular ${\rm SO}(n)$-invariant special
Lagrangian extension of~$A$ agrees with one of these~$n$
extensions in some open neighborhood of~$A$.
The method employed is an analysis of a singular nonlinear
\textsc{pde} and ultimately calls on the work of G\'erard and
Tahara to prove the existence of the extension. |
Keywords： |
Calibrations, Special Lagrangian submanifolds |
Classification： |
53C42, 35A20 |
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